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The three edge integrals of $z^{2}$ around the triangle with vertices $0$ , $1$ , and $i$ sum to zero

Example

For the positively oriented boundary of $Δ [0, 1, i]$ ,

$\int_{\partial Δ [0, 1, i]} z^{2} d z = 0.$

More precisely, the integrals along the directed edges $0 \to 1$ , $1 \to i$ , and $i \to 0$ are respectively

$\frac{1}{3}, \frac{- i - 1}{3}, \frac{i}{3} .$

Facts & Assumptions

Given: The oriented triangle $Δ [0, 1, i]$ and the integrand $f (z) = z^{2}$ .

[L1]

The oriented triangle boundary follows the directed edges $0 \to 1 \to i \to 0$ (Filled complex triangles, their oriented three-edge boundary contours, diameter, and perimeter).

[L2]

Complex polynomials are entire and obey the power derivative rule (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero).

[L3]

If $F$ is holomorphic with continuous derivative on an open set containing a rectifiable contour from $u$ to $v$ , then $\int F^{'} = F (v) - F (u)$ (The line integral of a continuous function admitting a primitive is that primitive's endpoint increment along every rectifiable path).

[L4]

Goursat's theorem gives zero integral for a holomorphic function around every filled triangle contained in its domain (Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain).

[L5]

A holomorphic function is continuous (Complex differentiability at a point implies continuity there).

Verification

technique · direct

1.1

The polynomial $F (z) = z^{3} / 3$ is entire with $F^{'} (z) = z^{2}$ ; this derivative is holomorphic by [L2] and continuous by [L5], so every hypothesis of [L3] holds on each edge.

L2L3L5

2.1

In the orientation of [L1], the endpoint increments are $F (1) - F (0) = 1 / 3$ , $F (i) - F (1) = (- i - 1) / 3$ , and $F (0) - F (i) = i / 3$ .

step 1.1L1L3algebra

3.1

Their sum is zero, proving the displayed integral directly; [L4] gives the same value because $z^{2}$ is entire.

step 2.1L2L4algebra∎

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The three edge integrals of z2z2 around the triangle with vertices 00, 11, and ii sum to zero